I have a pretty ferocious integral to solve, and since it doesn't seem I'll be able to do much analytically, I've taken to Mathematica to get some information. Mainly, I want to see if there are any constant terms in the following integral:

$$ I = \int_{r}^{\infty} r_0^{-5/2} W_{-i\alpha'/2, \nu}\left(\frac{-ir_0^2\omega'}{L}\right) W_{-i\alpha/2, \nu}\left(\frac{-ir_0^2\omega}{L}\right) J_{\nu}\left(-i\hat{k}r_0\right) dr_0$$

Where $W$ is WhittakerW, $J$ is BesselJ, and everything besides $r/r_0$ is a constant.

So far, I've simply been doing NIntegrate and breaking the integral up over several pieces. I plotted the function on a log plot, and it seems after about $r \approx 10$ it drops off significantly; as such, the integral from 10 to infinity seems to be zero. After this, I've just plotted the integral and tried to look at dependence using LogPlots. Does anyone have a better method to deal with Numerical Integrals like this to get a bit more information?

Here is my mathematica code:

g[r_, \[Alpha]_, v_, w_] := WhittakerW[-I*\[Alpha]/2, v, -I*r^2*w]

Plot[Log[(Re/Im)[
 r^(-5/2)*g[r, 10, 2.1, .01 ]*g[r, 15, 2.1, .01]*
BesselJ[3.2, 5*r]]], {r, 1, 100}, PlotLabel -> "Log(I2) vs r0"]

 % see that the function being integrated goes to zero for both Real and Imaginary parts
 

 I2bh[r_? NumericQ, v_] := 
 Re[NIntegrate[
 r0^(-5/2)*g[r0, 10, 2.1, .01 ]*g[r0, 15, 2.1, .01]*
  BesselJ[3.2, 5*r0], {r0, r, 15}, WorkingPrecision -> 15]]

I've done some more plotting and probing of the features of the integral, but for the purposes of this question I suppose what I have here is all I'm asking about